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Critical extinction exponents for a fast diffusion equation with nonlocal source and absorption

Haixia Li and Yuzhu Han*

Author Affiliations

Department of Mathematics, Jilin University, Changchun, 130012, P.R. China

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Boundary Value Problems 2014, 2014:24  doi:10.1186/1687-2770-2014-24

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/24


Received:22 August 2013
Accepted:10 January 2014
Published:30 January 2014

© 2014 Li and Han; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, the authors apply the super-solution and sub-solution methods, instead of energy estimate methods, to investigate the critical extinction exponents for a fast diffusion equation with a nonlocal source and an absorption term. They give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which improve, in some sense, the results by Xu et al. (Bound. Value Probl. 2013:24 2013) and by Han et al. (Arch. Math. 97:353-363, 2011).

MSC: 35K55, 35K57.

Keywords:
fast diffusion equation; critical exponents; extinction; super-solutions and sub-solutions

1 Introduction

In this paper, we investigate the following fast diffusion equation with a nonlocal source and an absorption term:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M3">View MathML</a>, Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M5">View MathML</a>) with smooth boundary Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M6">View MathML</a> is a nonnegative nontrivial function.

The equation in (1.1) is a fast diffusion equation perturbed by both a nonlocal source term and an absorption term, which describes the diffusion of concentration of some Newtonian fluids or the density of some biological species (see [1,2] and the references therein). What we are interested in here is the extinction in finite time of the nonnegative solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> of (1.1), i.e. there exists a finite time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8">View MathML</a> such that the solution is nontrivial for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M9">View MathML</a>, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M10">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M11">View MathML</a>. In this case, T is called the extinction time. As one of the most important properties of solutions of evolutionary equations, extinction in finite time of solutions has been intensively studied by several authors (see [3-16] and the references therein). In particular, in a recent paper by Xu et al.[13], the authors investigated the extinction and non-extinction phenomena of solutions of Problem (1.1) and obtained the following result.

Theorem 1.1 (Theorems 1-3 in [13])

Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M5">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M13">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M14">View MathML</a>, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time for any nonnegative initial data provided that either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M15">View MathML</a>orais sufficiently small; If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M13">View MathML</a>, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M15">View MathML</a>orais sufficiently small, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M19">View MathML</a>withCbeing a positive constant depending only onN, randm; If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M13">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M21">View MathML</a>, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time for any nonnegative initial data provided thatbis sufficiently large.

It can be seen from the above theorem that the extinction of nonnegative nontrivial weak solutions to Problem (1.1) occurs when the absorption term is in some sense strong. However, when the absorption term is suitably weak, whether Problem (1.1) admits non-extinction solutions or not is not answered in [13]. On the other hand, it can be seen from [9] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22">View MathML</a> is the critical extinction exponent when there is no absorption term. An interesting problem is whether the absorption term can change the critical extinction exponent. We know from a recent paper [17] by Liu et al. that the critical extinction exponent is not changed (at least when the source is local) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M23">View MathML</a>. However, when the absorption term is nonlinear, i.e. when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M24">View MathML</a>, the problem is open.

Motivated by the works mentioned above, we investigate the critical extinction exponents for Problem (1.1) by constructing suitable super and sub-solutions and give a more complete classification of exponents and coefficients for the solutions to vanish in finite time or not.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M25">View MathML</a> be the unique positive solution of the following linear elliptic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M26">View MathML</a>

(1.2)

Throughout this paper, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M27">View MathML</a>

(1.3)

By the strong maximum principle we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M28">View MathML</a>. Our main results are the following theorems.

Theorem 1.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M29">View MathML</a>, then all the solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>vanish in finite time for suitably small initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M33">View MathML</a>, then all the solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>vanish in finite time for any nonnegative bounded initial data.

Theorem 1.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M35">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M36">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M37">View MathML</a>, then all the solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>of Problem (1.1) vanish in finite time for appropriately small initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31">View MathML</a>.

Theorem 1.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M40">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M41">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M42">View MathML</a>, then Problem (1.1) admits at least one non-extinction solution for any nonnegative initial data.

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M43">View MathML</a> will be given in the proof of Theorem 1.4.

Theorem 1.5If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M44">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M45">View MathML</a>, then Problem (1.1) admits at least one non-extinction solution for any nonnegative initial data.

Theorem 1.6If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M46">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M47">View MathML</a>, then (1.1) admits at least one non-extinction solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>for any strictly positive initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M50">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M47">View MathML</a>, then Problem (1.1) admits at least one extinction solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>for any nonnegative initial data. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M50">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M54">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>vanishes in the sense that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M56">View MathML</a>.

Remark 1.1 Comparing Theorems 1.2-1.6 with Theorem 1.1 we can see that our results complement those obtained in [13] since the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M57">View MathML</a> is also considered in our paper. Moreover, according to Theorems 1.2, 1.5 and the first part of Theorem 1.6 it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22">View MathML</a> is the critical extinction exponent for Problem (1.1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M23">View MathML</a>, which is the same as the problems with local reaction terms (see [17]). However, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22">View MathML</a>, for the nonlocal problem under consideration, the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M62">View MathML</a> of −Δ in Ω no longer plays the same role as it does in the local case.

2 Proof of the main results

It is well known that the equation in (1.1) is degenerate if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M63">View MathML</a> and singular if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M2">View MathML</a>, and therefore there is no classical solution in general. To state the definition of the weak solution, we first define the class of nonnegative testing functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M65">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M66">View MathML</a>.

Definition 2.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M67">View MathML</a> is called a sub-solution (super-solution) of Problem (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68">View MathML</a> if the following conditions hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M69">View MathML</a> in Ω,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M70">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M71">View MathML</a>,

(iii) for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M72">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M73">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M74">View MathML</a>

(2.1)

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> is called a local solution of (1.1) if it is both a sub-solution and a super-solution for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> is called a solution of (1.1) if it is a local solution of (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8">View MathML</a>.

Local existence of weak solutions of (1.1) can be obtained by utilizing the methods of standard regularization (see [9]) and the continuity of the solutions can be derived by the arguments similar to that in [18]. Moreover, Problem (1.1) admits global solutions when the initial data are small (see [1]). Since the regularization procedure is crucial in what follows, we shall sketch the outline. Consider the regularized problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M80">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8">View MathML</a> may be chosen sufficiently small in such a way that there exists a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M82">View MathML</a> of (2.2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M84">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M85">View MathML</a> is bounded independently of k. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M86">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M87">View MathML</a>, and a super-solution (sub-solution) comparison theory holds for (2.2) (see [1,19]).

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M82">View MathML</a> is monotone in k, we may define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M89">View MathML</a>, and it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M90">View MathML</a> is a solution of (1.1). Furthermore, if u is a solution of (1.1), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M91">View MathML</a>

where we use the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M92">View MathML</a> on Ω to derive this inequality. With <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M94">View MathML</a> defined so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M95">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M96">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M97">View MathML</a>

Thus, we can choose the appropriate test function ξ as in [1,19] to obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M98">View MathML</a>. If u is a sub-solution of (1.1), the above argument shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M98">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M90">View MathML</a> is the maximal solution of (1.1), and this solution satisfies a sub-solution comparison principle.

Before proving our main results, we give a comparison principle for the solution of Problem (1.1), which is similar to Proposition 2.3 in [9] and can be proved by modifying the above arguments (see also [1,10,19]).

Proposition 2.1Letuandvbe a nonnegative bounded sub-solution and a nonnegative super-solution of (1.1), respectively. If either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M101">View MathML</a>anduis bounded from the above or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M102">View MathML</a>andvhas a positive lower bound, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M103">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M105">View MathML</a>in Ω.

Proof of Theorem 1.2 Case (i): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M33">View MathML</a>. For any bounded smooth domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M108">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M109">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M110">View MathML</a> be the unique solution of the following elliptic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M111">View MathML</a>

(2.3)

By the comparison principle for linear elliptic problem we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M112">View MathML</a> in Ω. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M115">View MathML</a>. It is well known from the strong maximum principle that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M116">View MathML</a>.

By continuity, we can choose a suitable domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M108">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M118">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M119">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M120">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M122">View MathML</a>

(2.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M2">View MathML</a>, it follows from the theory in ODEs that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> is nonincreasing and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M125">View MathML</a> for all

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M126">View MathML</a>

Then it can be verified that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> is a super-solution of (1.1). In fact, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M129">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> satisfies the following inequalities (in the weak sense):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M131">View MathML</a>

(2.5)

In addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M132">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M71">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M134">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M135">View MathML</a> by the choice of A. Moreover, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M136">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M137">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68">View MathML</a>. Therefore, by applying Proposition 2.1 to (1.1) we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M103">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M140">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M141">View MathML</a>. The arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M143">View MathML</a> ensure that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M144">View MathML</a>. Furthermore, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M145">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M146">View MathML</a> satisfies (1.1) with the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M147">View MathML</a>. By the aforementioned proof, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M148">View MathML</a> with any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M149">View MathML</a>. From the relation of the extinction time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M150">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> to A, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M152">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M153">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M154">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M155">View MathML</a>.

Case (ii): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M29">View MathML</a>. Let ϕ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M157">View MathML</a> be the same as Case (i) and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M158">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M159">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M160">View MathML</a>, then it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> is a super-solution of (1.1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31">View MathML</a> is sufficiently small such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M163">View MathML</a> in Ω. Applying Proposition 2.1 to Problem (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M8">View MathML</a> we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M103">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M68">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M168">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M170">View MathML</a>

By the choice of k and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M171">View MathML</a> it is easily verified that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M172">View MathML</a>. Thus, by the results of Case (i), we can conclude that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> vanishes in finite time when the initial data are suitably small. The proof of this theorem is complete. □

Proof of Theorem 1.3 We first prove the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M36">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M175">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M176">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> satisfies the following ordinary differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M178">View MathML</a>

(2.6)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M175">View MathML</a>, we know by integrating the ODE that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> vanishes at some finite time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M150">View MathML</a>. Moreover, as in the proof of Theorem 1.2, it can be verified that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> is a super-solution of (1.1). Thus, by applying Proposition 2.1 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M134">View MathML</a> we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> also vanishes at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M150">View MathML</a>.

In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M35">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> satisfy the following ODE:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M191">View MathML</a>

(2.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M192">View MathML</a>. Similar to the first case, it is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> vanishes in finite time since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> is a super-solution of (1.1) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M31">View MathML</a> is small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M197">View MathML</a>. Applying Proposition 2.1 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> guarantees the finite time extinction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>. This completes the proof of Theorem 1.3. □

Proof of Theorem 1.4 (i) Consider first the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M201">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M62">View MathML</a> be the first eigenvalue of the following eigenvalue problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M203">View MathML</a>

(2.8)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M204">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M205">View MathML</a>) be the corresponding eigenfunction. We may normalize <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M206">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M207">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M208">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> satisfy the ODE problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M210">View MathML</a>

(2.9)

It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> is nondecreasing and bounded from above by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M212">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M213">View MathML</a>. We shall show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> is a sub-solution of (1.1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M215">View MathML</a> is sufficiently small. In fact, simple computations show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M216">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M217">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> to be a sub-solution of (1.1), it suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M219">View MathML</a>

which follows from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M220">View MathML</a>

(2.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M221">View MathML</a>. It is easy to see that (2.10) is valid for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M215">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M223">View MathML</a>.

Next, we turn our attention to construct a super-solution of (1.1). Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M224">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M225">View MathML</a>. Then it is not hard to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226">View MathML</a> is a super-solution and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M227">View MathML</a>. Therefore, by an iteration process, one can obtain a solution of Problem (1.1), which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M228">View MathML</a>. Indeed, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M229">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M230">View MathML</a> iteratively to be a solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M231">View MathML</a>

subject to the boundary and initial conditions as that in (1.1). By applying the comparison technique used in the proof of Lemma 2.1 in [1,12] we know that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M232">View MathML</a>, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M233">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M153">View MathML</a>, is a solution of (1.1). Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> does not vanish, neither does <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>.

(ii) The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M237">View MathML</a> can be treated similarly to Case (i).

(iii) Finally we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M41">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M42">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> satisfy the following ODE:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M241">View MathML</a>

(2.11)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> is nondecreasing and satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M243">View MathML</a>. (The upper bound of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> can be obtained by contradiction arguments and the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> follows immediately as the upper bound is derived.) As in the proof of Case (i), we can construct a non-extinction sub-solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M246">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M215">View MathML</a> sufficiently small.

To construct a super-solution, consider the following eigenvalue problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M248">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M249">View MathML</a> is a bounded domain with smooth boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M250">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M252">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M253">View MathML</a>) be its first eigenvalue and the corresponding eigenfunction, respectively. We may normalize <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M254">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M255">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M256">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M257">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M258">View MathML</a>, then we shall show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226">View MathML</a> is a super-solution of (1.1) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M260">View MathML</a> is suitably large. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M261">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M262">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M264">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226">View MathML</a> satisfies the following inequalities (in the weak sense):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M266">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M227">View MathML</a> by the choice of k. Therefore, by applying the monotonicity iteration process we can obtain a non-extinction solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> of (1.1) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M228">View MathML</a>. The proof of Theorem 1.4 is complete. □

Proof of Theorem 1.5 The proof of this theorem is similar to that of Theorem 1.4, so we only sketch the outline here. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M270">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M25">View MathML</a> is defined in (1.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> satisfies the following ODE problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M273">View MathML</a>

(2.12)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M275">View MathML</a>, it is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> is nondecreasing and bounded above by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M277">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> is a sub-solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M215">View MathML</a> is sufficiently small. On the other hand, the super-solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226">View MathML</a> can be chosen to be a large positive constant L satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M281">View MathML</a>. It can be observed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M282">View MathML</a> is a pair of sub-solution and super-solution of (1.1) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M227">View MathML</a>. Therefore, by monotonicity iteration, we know that (1.1) admits at least one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M228">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M286">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> cannot vanish at any finite time. The proof of Theorem 1.5 is complete. □

Proof of Theorem 1.6 (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> be any solution of (1.1). It can be verified that, for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M290">View MathML</a>, a sufficiently large constant L is a super-solution of (1.1). Therefore, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M291">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M292">View MathML</a>. For convenience, in the following proof, we assume that the weak solution is appropriately smooth; otherwise, we can consider the corresponding regularized problem, and the same result can also be obtained through an approximate process (see [15]). Multiplying equation (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M293">View MathML</a> and integrating by parts over Ω yield the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M294">View MathML</a>

(2.13)

Recall the embedding theorem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M295">View MathML</a>

Combining this result with (2.13) and using Hölder’s inequality on the right hand side of (2.13) one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M296">View MathML</a>

(2.14)

Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M291">View MathML</a>, we see from (2.14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M299">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M300">View MathML</a>

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M301">View MathML</a> tends to 0 exponentially as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M302">View MathML</a>.

(ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M303">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M305">View MathML</a>

(2.15)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M121">View MathML</a> is nonincreasing and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M125">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M309">View MathML</a>. Noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M47">View MathML</a>, one can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M226">View MathML</a> is a super-solution of (1.1) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M313">View MathML</a> in Ω. By using the arguments similar to that of the proof of Case (i) of Theorem 1.2 we can show that any solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a> of Problem (1.1) vanishes in finite time.

(iii) Finally we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M46">View MathML</a>. First we construct a non-extinction sub-solution of (1.1). Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M316">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M317">View MathML</a>, α are two positive constants to be determined. Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M47">View MathML</a>, it is easily verified that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> is a sub-solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M321">View MathML</a> and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M317">View MathML</a> is so small such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M323">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M324">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> to be sub-solution of (1.1) it is reasonable to choose first <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M317">View MathML</a> so small such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M323">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M328">View MathML</a>. Next, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M329">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> is bounded, we can choose a sufficiently large constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M331">View MathML</a> to be a sup-solution of (1.1). Therefore, by monotonicity iteration, we can obtain a solution of (1.1) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M332">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M127">View MathML</a> does not vanish at any finite time, neither does <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/24/mathml/M7">View MathML</a>. The proof of Theorem 1.6 is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which improve the original manuscript. The authors would also like to express their sincere gratitude to Professor Wenjie Gao for his enthusiastic guidance and constant encouragement. The project was supported by NSFC (11271154).

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